The number of one-one functions $f : \{1, 2, 3, 4, 5\} \to \{0, 1, 2, 3, 4, 5\}$ such that $f(1) \neq 0, 1$.

Question

This question was asked in the ISI PGDSMA entrance exam. Can anyone help me in this? I am getting $480$ as answer.

Answer 1

There are a total of $\binom{6}{5} 5!=6!$ injections from a five element set to a six element set.

We count the number of injections where $f(1)=0$ and $f(1)=1$, then use the addition principle of counting.

Case 1: $f(1)=0$.

In this case, there are $\binom{5}{4}4!= 5!$ total injections with $f(1)=0$.

Case 2: $f(1)=0$

This is the same as case one and there are $5!$ injections of this kind.

Now there are $$6!-2\cdot 5!= 480$$ injections where $f(1)\neq 0$ or $f(1)\neq 1$.

Your answer was spot o$n!$